Review of AdS/CFT Integrability, Chapter III.1: Bethe Ans\"atze and the R-Matrix Formalism

TL;DR

This paper reviews the integrability of the Heisenberg XXX spin chain within the AdS/CFT framework, discussing solution methods like Bethe ansatz and R-matrix formalism, and exploring potential extensions to the exact AdS/CFT solution.

Contribution

It provides a comprehensive overview of Bethe ansatz techniques and proposes a novel approach to construct the Baxter Q-operator, potentially simplifying spectrum calculations.

Findings

Detailed comparison of coordinate and algebraic Bethe ansatz methods

Discussion on lifting integrability techniques to the AdS/CFT system

Proposal of a new method for constructing the Baxter Q-operator

Abstract

The one-dimensional Heisenberg XXX spin chain appears in a special limit of the AdS/CFT integrable system. We review various ways of proving its integrability, and discuss the associated methods of solution. In particular, we outline the coordinate and the algebraic Bethe ansatz, giving reference to literature suitable for learning these techniques. Finally we speculate which of the methods might lift to the exact solution of the AdS/CFT system, and sketch a promising method for constructing the Baxter Q-operator of the XXX chain. It allows to find the spectrum of the model using certain algebraic techniques, while entirely avoiding Bethe's ansatz.